Grinblatt & Liu 2008 - Debt Policy, Corporate Taxes, And Discount Rates (JET)

Journal of Economic Theory 141 (2008) 225–254www.elsevier.com/locate/jet

Debt policy, corporate taxes, and discount rates

Mark Grinblatta, Jun Liub,c,d,∗,1

aUCLA Anderson School of Management, USAbCheung Kong Graduate School of Business, China

cSouthwestern University of Finance and Economics, ChinadRady School at UCSD, USA

Received 19 October 2004; final version received 7 November 2006; accepted 12 September 2007Available online 22 October 2007

Abstract

This paper applies the standard risk-neutral valuation framework to tax shields generated by dynamicdebt policies. We derive a partial differential equation (PDE) for the value of the debt tax shield. For aclass of dynamic debt policies that depend on the asset’s free cash flows, value, and past performance, weobtain closed-form solutions for the PDE. We also derive the tax-adjusted cost of capital for free cash flowsand analyze the conditions under which the weighted average cost of capital is an appropriate discountrate. Finally, we derive closed-form solutions for equity betas, which differ from the formulas that havetraditionally been used to lever and unlever equity betas.© 2007 Elsevier Inc. All rights reserved.

JEL classification: G1; G3

Keywords: WACC; Discount rate; Tax shields; Corporate tax

1. Introduction

An intrinsic difficulty associated with the valuation of debt tax shields is identifying the riskof the tax deductions arising from the stream of future debt interest expenses. As a consequence,the rate at which one discounts the future stream of interest-related tax shields, and hence thevalue of those tax shields, has eluded prior research, except for the simplest of cases. These casesimpose stringent restrictions on the cash flow process and debt policy to circumvent the complexissue of risk and valuation. Among these are the models of [17,16].

∗ Corresponding author. Rady School at UCSD, USA. Fax: +1 858 534 0745.E-mail address: [email protected] (J. Liu).

1 On leave from the Rady School at UCSD, USA.

0022-0531/$ - see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2007.09.009

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226 M. Grinblatt, J. Liu / Journal of Economic Theory 141 (2008) 225–254

The Modigliani and Miller (M&M) debt policy is one where the debt level is constant, perpetual,and not subject to default (although the formulas derived by M&M extend as is to the case wheredefault always leads to zero recovery). This debt policy implies that one can discount the stream offuture interest-based tax shields at the risk-free rate. If the tax rate is constant, as they assume, thedebt tax shield’s present value is necessarily proportional to the present value of the debt becausethe cash flow stream from the tax shield is proportional to the cash flow stream from the debt.Here, since the constant of proportionality is the corporate tax rate, the present value of the debttax shield is the product of the corporate tax rate and the present value of the debt. Modigliani andMiller [17] also use this model to develop formulas for discount rates that account for the valueof the tax shield when cash flows have no tendency to grow.

The interesting case studied by Miles and Ezzell focuses on the dynamic issuance of perpetualrisk-free debt. This case assumes: (1) the free (or unlevered) cash flow (the after-tax cash flowthat stems directly from the real asset, which is unaffected by the financing mix) follows a randomwalk with no drift; (2) the cash flow stream is valued by applying a constant discount rate; (3)the debt-to-asset ratio is constant; and (4) debt has no default. Under these assumptions, the cashflow from each date’s tax shield is of the same risk as the one period lagged free cash flow. AsGrinblatt and Titman [9], and Brealey and Myers [2] point out, in the continuous time limit ofthis model, the cash flow stream from future debt-related tax deductions is of the same risk asthe free cash flow stream and thus can be discounted at the same rate. Miles and Ezzell, as wellas the standard textbooks, present formulas analogous to those in Modigliani and Miller for thisalternative debt policy.

The actual debt policies of firms tend to deviate from those specified by the Modigliani–Millerand Miles–Ezzell models. 2 In these cases, the literature in finance offers little guidance onvaluation. For example, when debt interest is nonlinear in the free cash flow, the value of the debttax shield and the specification of the after-tax discount rate for valuing the asset are not known.

This paper proposes a no-arbitrage valuation of debt tax shields for a class of Markovian debtpolicies. The value of the debt tax shield for dynamic debt policies can be derived as the solutionto a partial differential equation (PDE). Our approach allows for stochastic cash flow, stochasticdiscount rates, and realistic recovery assumptions on default. For a large class of dynamic debtpolicies, we can reduce the PDE to a system of ordinary differential equations, which are easilysolved numerically. With more restrictive assumptions, but still with more generality than existsin the literature, we obtain the value of the tax shield as a closed-form expression.

The discount rate for free cash flows that accounts for the debt tax shield is also of criticalimportance, both to practitioners and researchers. We show what adjustments are needed to convertthe weighted average cost of capital (WACC) to an appropriate discount rate. Such adjustments arealmost always needed as the WACC is an appropriate discount rate only in the Modigliani–Millerand Miles–Ezzell cases, or in some linear hybrid of these two well-known cases.

Finally, for general debt policies, we derive a formula that characterizes the equity beta as afunction of the leverage ratio and the unlevered asset beta. This formula’s special cases includethe standard textbook formulas of Hamada [10] and Miles and Ezzell [16], which are associatedwith the Modigliani–Miller and Miles–Ezzell models, respectively.

Our approach differs from that found in prior research on debt tax shield valuation. In lieu ofstrong restrictions on cash flows, discount rates, debt policy, or recovery on default, we use therisk-neutral pricing framework in a continuous-time setting, following the option pricing approachof Black–Scholes [21] and Merton [22], and the insights of Cox and Ross [3], Ross [18], and

2 See, for example, Donaldson [4], Baker and Wurgler [1], Graham and Harvey [7], and Kayhan and Titman [13].

M. Grinblatt, J. Liu / Journal of Economic Theory 141 (2008) 225–254 227

Harrison and Kreps [11]. The risk-neutral valuation methodology of asset pricing theory has beenapplied before in a corporate setting, perhaps most elegantly by Ross [18]. Our approach, whilesimilar in spirit, uses this methodology to derive solutions that apply specifically to the debt taxshield. Moreover, our closed-form solutions yield economic insights and intuition about the valueof the debt tax shield. One example of this is a heuristic description of how the value of the debttax shield, as well as the appropriate discount rate, vary with debt policies that can be viewed asweighted averages of the Modigliani–Miller and Miles–Ezzell debt policies.

The goal of our paper is to assess whether broad classes of exogenously specified debt poli-cies lend themselves to tractable valuation solutions. The optimal debt policy, while importantfor deciding which of our debt policies should be focussed on, is beyond the scope of this pa-per. The optimal state-contingent debt path is a considerably different problem, and a generallyintractable one, once the economic framework becomes slightly less restrictive than that in theextant literature. 3

Section 2 of the paper develops a general approach for valuing debt tax shields. It also analyzesa large class of debt policies which have closed-form solutions for the tax shield and presentstwo larger classes of cases for which numerical computation of the value of the debt tax shield istrivial. Section 3 examines the weighted average cost of capital and relates it to valuation. It alsocharacterizes how the WACC is affected by dynamic debt policies and studies when the WACCcan be used to obtain valuations that properly account for the value of the debt tax shield. Finally,this section derives closed-form solutions for tax-adjusted discount rates that generate the correcttax-adjusted valuations of free cash flow streams. In most cases outside of the Modigliani–Millerand Miles–Ezzell frameworks, we show that these discount rates differ from the WACC. Section 4analyzes how to lever and unlever equity betas and equity risk premia for arbitrary debt policies.Section 5 concludes the paper.

2. The valuation of debt tax shields

In a dynamically complete market, two assets with payoffs driven by the same source ofuncertainty are instantaneously perfectly correlated. This allows them to be valued in relation toone another. Just as an option (whether listed or synthetically created from a dynamic strategy)is valued in relation to its underlying security, so too can a debt tax shield be valued in relation tothe unlevered asset it is associated with.

In the continuous-time framework, it is irrelevant that the unlevered asset may not be directlytraded as a security. Just as we know the no-arbitrage value one should place on a synthetic optionthat is created from a dynamic portfolio strategy in a stock and a bond, we know the value of whateffectively is a levered asset (and hence a tax shield) as a function of the value of an otherwiseidentical unlevered asset, and vice versa. So long as the uncertainty behind the debt policy thatgenerates the tax shield is tied only to uncertainty in the free cash flows, debt tax shields aresimply derivatives. For this reason, most of the paper assumes that the free cash flow, the after-taxcash flow that would be generated in the absence of debt financing, satisfies a general Markovdiffusion process.

In our framework, it also makes no difference whether we treat the unlevered asset or the leveredasset or the tax shield as the underlying security. We are simply valuing these cash flow claims in

3 Optimal debt levels when the bankruptcy costs of debt are weighed against tax benefits and costs are found in Leland[15], Green and Hollified [8], and Ross [19].


relation to one another. For expositional convenience, and economic intuition that is consistentwith the extant literature, we use the unlevered asset as the more primitive underlying security.

An asset that is levered with risk-free debt has two sources of after-tax cash flow at date t: (1)the free (or unlevered) cash flow, Xtdt , which is unaffected by the asset’s financing mix; and (2)the flow from the debt interest tax shield, �cDt rf dt , which is the product of the tax rate �c andthe debt interest payment Dtrf dt . 4 Because valuation is linear, the value of a levered asset is thesum of the values of its two cash flow components.

For simplicity (and without loss of generality), we let one Brownian motion, B, drive theuncertainty. That is, between dates 0 and T, with T possibly infinite,

dXt = g(t, X)X dt + �(t, X)X dBt .

With this assumption, the market is dynamically complete with two tradable assets. 5 This meansthat a dynamic trading strategy can transform a levered asset into an unlevered asset, and viceversa. Similarly, knowing the value of an asset for any given debt policy allows us to compute thevalue of its tax shield for all debt policies. Solutions can be found with a variety of mathematicallyequivalent approaches, but the most popular method involves the solution of a differential equationgenerated by Ito’s Lemma and the principle of no arbitrage.

The continuous-time setting, described above, allows valuation of almost any derivative, in-cluding tax shields, by applying the well-known no-arbitrage principle. For expositional clarity,the model initially assumes no default on debt and that the debt is short-term. We also assume thatthere are no personal taxes or other market frictions associated with debt beyond the corporatetax. 6 One can easily extend the model to account for these kinds of frictions, as we will discussshortly.

2.1. Debt policy and levered asset valuation: the PDE

We begin by studying the valuation of a levered asset under a general class of debt policies.The debt policy D can be any differentiable function of time t, free cash flow X, levered assetvalue V L, and history H. That is,

Dt = D(t, Xt , VLt , Ht ).

These general debt policies can depend, in a quite complicated manner, on the history of the asset,such as past cash flows, past debt values, past asset values, in addition to current cash flow and

4 In order for the levered asset to have the same investment policy in the presence of debt, we assume, without loss ofgenerality, that the flow from the interest-based tax shield is paid out. It could be retained in a risk-free interest bearingaccount and distributed later, but this has tax consequences for the firm. In essence, such retention amounts to negativedebt and it is the net debt policy for which we are computing the tax shield. Given this definition of how to accountfor debt, and appropriate care taken to avoid double counting when this cash is eventually distributed, our results applyirrespective of whether cash is retained or paid out. Similarly, our valuation equations appear to assume that free cash flowis distributed to debt and equity holders. However, except for the tax consequences of the firm’s unnecessary retention ofliquid assets, as described above and which is easily adjusted for, the retention of future free cash flows does not affect anasset’s present value.

5 This can be a risky and a risk-free security or two risky securities with uncertainty driven by Bt .6 Personal taxes are clearly important for asset valuation and debt policy, as Green and Hollifield [8] prove theoretically

and document empirically. Their paper analyzes the optimal capital structure for a firm with a Modigliani–Miller debtpolicy, bankruptcy costs, and a cash distribution policy to equity holders that is sensitive to the economic effects of thecorporate and capital gains taxes. Ross [19] studies the cost of capital and optimal capital structure with both corporateand personal taxes in a setting with constant debt payments in no-default states.


current asset value. 7 We only require that the history dependence at a given date t be summarizedby additional date t state variables. This allows us to maintain the Markovian setting. Note, too,that debt policy depends on the value of the levered asset, which in turn will depend on debtpolicy. However, our valuation approach simultaneously determines the value of both the debt taxshield and the value of the levered asset. (This is also true, for example, in Miles and Ezzell.)

Without loss of generality, we simplify notation by treating these state variables as the singlevariable Ht . 8 As long as the uncertainty associated with the path of X spans the relevant statespace for H, we will still be able to value the debt tax shield as a function of the value of theunlevered asset. To maintain this desirable property, we assume that Ht satisfies the diffusion

dHt = �H (t, Xt , Dt , VLt ) dt + �H (t, Xt , Dt , V

Lt ) dXt

= �h(t, Xt , Dt , VLt ) dt + �h(t, Xt , Dt , V

Lt ) dBt ,

where

�h(t, Xt , Dt , VLt ) = �H (t, Xt , Dt , V

Lt ) + �H (t, Xt , Dt , V

Lt )g(t, Xt )Xt

and

�h(t, Xt , Dt , VLt ) = �H (t, Xt , Dt , V

Lt )�(t, Xt )Xt .

Recognize that the functional form of the exogenously specified �h and �h can be quite general,which allows us to analyze many empirically relevant debt policies.

Given this description of debt policy and history, it follows that the date t value of the leveredasset, V L

t = V L(t, Xt , Ht ), depends on the current date, cash flow, and history. Moreover, if thefree cash flow stream terminates at date T (essentially, becomes 0 at date T and forever thereafter),the functional form of the valuation function will be influenced by the proximity to the terminationdate.

In the absence of arbitrage, dynamic completeness implies that this value necessarily satisfiesthe partial differential equation (PDE)

�V L

�t+ (g − �)X

�V L

�X+ 1

2�2X2 �2

V L

�X2

+(�h − ��h)�V L

�H+ 1

2�2

h

�2V L

�H 2+ ��h �2

V L

�H�XX = rf V L − (X + rf �cD), (1)

where we have dropped the arguments of g, �, etc., for notational simplicity. If the asset has afinite life, the terminal condition is

V L(T , XT , HT ) = 0.

This partial differential equation, a familiar extension of the well-known Black–Scholes differ-ential equation, is simply the no-arbitrage condition associated with an asset whose uncertainty isspanned by the payoff to a dynamic trading strategy in the unlevered asset and a risk-free security.The �(t, X) term in Eq. (1), (shortened to � for notational simplicity), is the premium per unitof risk generated by changes in B. In a corporate setting, it would be traditional to think of this

7 Kayhan and Titman [13] have documented the importance of history on leverage ratios.8 We can also regard H and the drift coefficients as vectors and the diffusion coefficients as a matrix with virtually no

change to any of our equations.


parameter as being determined by the instantaneous discount rate of the unlevered asset. However,� also can be inferred from the levered asset’s value (or levered equity value) for any debt policy.

To derive the partial differential equation, note that Ito’s Lemma implies that the change in thevalue of a levered asset plus all distributions of cash flow:

dV Lt + (Xt + rf �cDt ) dt

= �V L

�tdt + �V L

�XdX + �V L

�HdH

+(

1

2

�2V L

�X2�2X2 + �2

V L

�H�X��h + 1

2

�2V L

�H 2�2

h

)dt + (Xt + rf �cDt ) dt

=(

�V L

�t+ �V L

�XgX + �V L

�H�h

)dt

+(

1

2

�2V L

�X2�2X2 + �2

V L

�H�X��hX + 1

2

�2V L

�H 2�2

h + Xt + rf �cDt

)dt

+(

�V L

�X+ �V L

�H�H

)�X dB.

The analogous equation for an otherwise identical unlevered asset with date t value V Ut =

V U(t, X) is

dV Ut + Xt dt =

(�V U

�t+ �V U

�XgX + 1

2

�2V U

�X2�2X2 + Xt

)dt + �V U

�XX� dB

=(

V Urf + �V U

�XX�

)dt + �V U

�XX� dB,

where �, defined by its placement above, is simply a convenient symbol for a scaling of the riskpremium attached to dB. We can express � in terms of the instantaneous unlevered cost of capital,rUt as

�(t, X) = rUt − rf

� ln(V Ut )

� ln(Xt )

. (2)

Clearly, instantaneous changes inV L andV U are perfectly correlated. Thus, to prevent arbitrage,the ratios of the risk premia per dollar invested in the levered and unlevered assets must beproportional to the risk born per dollar invested in each of the assets. This implies

�V L

�t+ �V L

�XgX + �V L

�H�h + 1

2

�2V L

�X2�2X2 + �2

V L

�H�X��hX + 1

2

�2V L

�H 2�2

h + Xt

+rf �cDt − rf VL =(

�V L

�X+ �V L

�H�H

)X�,

which, when rearranged, gives us Eq. (1). In analogous fashion, the value of the tax-shield � =V L − V U satisfies the PDE

��

�t+ (g − �)X

��

�X+ 1

2�2X2 ��

�X2

+(�h − ��H )��

�H+ 1

2�2

h

�2�

�H 2+ ��h

�2�

�H�XX = rf � − rf �cD. (3)


2.2. Discussion

The interpretation of Eq. (1) is straightforward. The left side represents the expected rate ofchange in the asset’s value under the risk-neutral measure. On the right side, the term in parenthesesis the sum of the free cash flow and the cash flow from the tax-shield. Thus, if we divide both sidesby VL, our PDE is simply the well-known result that in the absence of arbitrage, the asset valueunder the risk-neutral measure has an expected rate of growth equal to the difference between therisk free return and the dividend yield (from payouts of the free cash flow and the tax-shield).

As long as the short-term risk-free rate and the discount rate (which can be any function ofthe information set) for the cash flows of an otherwise identical unlevered asset are specified,the standard continuous-time valuation methodology prices any future cash flow contingent onthe same source of uncertainty. This includes the cash flow from the unlevered asset, the leveredasset, or a tax-shield from a complex, yet realistic, debt policy.

When markets are complete, the risk-neutral drift of the cash flow X is uniquely inferred fromthe price dynamics of any traded asset with uncertainty driven by Bt . The useful economic intuitionthat a portfolio of traded securities tracks the cash flow produced by the debt tax shield also applieshere. In such a complete markets setting, where we only try to capture relative values of claimsto levered and unlevered after-tax cash flows, there is no reason to specify what the traded assetsare. In principle, any financial instruments that have exposure to Bt can be used to determine theunique risk-neutral drift. As a practical matter, it is convenient to think of � as a function of therisk-premium on the unlevered asset, whether that asset is traded directly or created indirectlyfrom a dynamic strategic in a levered company’s equity and risk-free debt.

Even in the continuous-time setting, markets can be incomplete because debt policy is affectedby stochastic interest rates, stochastic volatility, industry structure and competition, macroeco-nomic conditions, and managerial discretion. When markets are incomplete, the valuation equa-tions shown above are still valid, but the risk-neutral drift for X in Eq. (1) is non-unique. 9 Hence,a violation of the pricing equation for a fixed set of risk-neutral drifts does not necessarily implyarbitrage. However, valuations that are not consistent with any choice of risk-neutral drifts wouldimply arbitrage, as the fundamental theorem of asset pricing proves.

Our analysis does not account for personal taxes in the equilibrium cost of debt to firms. Ross[19], in an elegant paper, considers both corporate and personal taxes. He uses a no-arbitrageapproach to value tax shields when firms can default. Furthermore, he derives the optimal capitalstructure from balancing the trade-off between the tax shield and bankruptcy costs. The debt taxshield conditional on no-default is a constant in Ross [19]. We have state-dependent debt taxshields, but do not consider bankruptcy costs, personal taxes, or optimal capital structure.

While the issue of optimal capital structure is beyond the scope of this paper, minor changes innotation on the right side of Eq. (1) easily address the bankruptcy cost and personal tax issues. Inthese more general cases, we need to replace rf in the term rf V L with the (potentially stochastic)default-adjusted short rate used in Duffie and Singleton [5] and the cash flow from the debt taxshield, rf �cD, with the (potentially state-contingent) effect of both personal and corporate taxeson the firm’s after-tax cost of debt in the absence of default. The derivation of the PDE, a non-trivial extension of the Duffie and Singleton [5] default model, is available from the authors uponrequest.

In principle, these partial differential equations (with or without extensions to bankruptcy andmore general tax issues) can be solved. However, without further restrictions, these differential

9 See Harrison and Kreps [11] for a discussion of this point in more general valuation settings.


equations are difficult to solve, even numerically. Hence, the remainder of this section explorescases where solutions are insightful or, from a numerical perspective, quickly attainable. Essen-tially, whenever we can transform the PDE into an ordinary differential equation (ODE), numericalsolutions are easily found. We explore two such classes of cases. In the first, all of the model’sparameters depend only on the contemporaneous level of the free cash flow, X. Here, because thereis no time dependence, Eq. (1) reduces to an ODE in X. In the second class of cases, which werefer to as “Additively Separable Assets,” the levered asset’s value is additively separable in a setof arguments, which consists of H and a finite collection of real powers of X: X�1 , X�2 , . . . , X�N .The coefficients of these arguments may be time dependent. The additively separable class ofcases is particularly interesting for its ability to generate closed-form solutions for the value of thedebt tax shield. These apply to both finite-lived and perpetual assets. They arise whenever the Ncash flow coefficients of the value additive functions for debt and history are growing at constantexponential rates and the remaining coefficients are constant. Given this level of generality, itappears as if our closed-form solutions could generate fairly good approximations for the valueof a debt tax shield for many interesting debt policies.

The closed-form solutions only require state-independent parameters for the cash flow process,the short rate, and the tax rate. For simplicity in notation, we present results for the case whereparameters of the cash flow process, the short rate, and the tax rate are constant.

2.3. Cash flows and debt with time-independent parameters

When the parameters of the dynamic process for X do not depend explicitly on time t (that is,g(t, X) = g(X), �(t, X) = �(X), and �(t, X) = �(X)) and the debt policy D does not dependon either time t or the history Ht , the asset value V L depends only on the contemporaneous cashflow level, X. In this case, the PDE for the value of the levered asset (1) becomes the second orderODE:

(g(X) − �(X))X�V L

�X+ 1

2�2(X)X2 �2

V L

�X2− rf V L = −(X + rf �cD(X)),

which is trivial to solve numerically for any specification ofg(X),�(X), and�(X). 10 Special caseswith closed-form solutions include the continuous-time versions of the Modigliani–Miller debtpolicy (g(X) = 0, �(X) = � and D(X) = D implying V U = X/(rf + �) and V L = V U + �cD)and the Miles–Ezzell debt policy (g(X) = 0, �(X) = � and D(X) = dxX, with dx constant,implying V U = X/(rf +�) and V L = V U +�cDrf /(rf +�)). We defer further discussion of thisas the class of debt policies analyzed next also includes the Modigliani–Miller and Miles–Ezzellmodels as special cases.

2.4. Additively separable assets: numerical solutions

Additively separable assets have tax shields with values that are additively separable linearfunctions of history, H, and any set of real powers of the cash flow, X�. Simple examples of

10 Any two boundary conditions, which implicitly determine the debt level in all states of the world, determine a uniquesolution to the differential equation. Hence, specifying the debt policy is clearly sufficient for obtaining the levered asset’svalue.


additively separable assets include the constant coefficient quadratic case,

V Lt = V U

t + cx0 + cx

1Xt + cx2X2

t ,

which is generated by the constant coefficient history-independent quadratic debt policy

Dt = dx0 + dx

1 Xt + dx2 X2

t

and the constant coefficient square root case,

V Lt = V U

t + cx0 + cx

1/2

√Xt,

which is generated by the constant coefficient history-independent square root debt policy

Dt = dx0 + dx

1/2

√Xt .

In Leland [15] and Ross [19], the optimal capital structure has a debt level that is a fractionalpower of the underlying cash flow.

A more complicated case arises when the debt level is history dependent with history given by

Ht = H0 e−mht− 12 (lh)2t−lhBt + mx

∫ t

0e−mh(t−s)− l

2 (lh)2(t−s)−lh(Bs−Bt )Xs ds

+md

∫ t

0e−mh(t−s)− 1

2 (lh)2(t−s)−lh(Bs−Bt )Ds ds

+mv

∫ t

0e−mh(t−s)− 1

2 (lh)2(t−s)−lh(Bs−Bt )V Ls ds

+lx∫ t

0e−mh(t−s)− 1

2 (lh)2(t−s)−lh(Bs−Bt )Xs dBs

+ld∫ t

0e−mh(t−s)− 1

2 (lh)2(t−s)−lh(Bs−Bt )Ds dBs

+lv∫ t

0e−mh(t−s)− 1

2 (lh)2(t−s)−lh(Bs−Bt )V Ls dBs.

In this special case, the diffusion process for Ht satisfies

dHt = (mxXt + mdDt + mvV Lt − mhHt) dt + (lxXt + ldDt + lvV L

t − lhHt ) dBt

and thus has drift and volatility of

�h = mxXt + mdDt + mvV Lt − mhHt , �h = lxXt + ldDt + lvV L

t − lhHt .

This history process, with the m’s and l’s constant, 11 when combined with an analogous functionalform for the debt process, leads to a closed-form additively separable solution for the value of thedebt tax shield, as we show in the next subsection.

11 The special case, md �= 0 while mx = mv = ld = lx = lv = 0, says that the debt depends on past debt levels, asempirical work seems to suggest.


The most general class of additively separable assets has history diffusion and debt policy ofthe form:

dHt =⎛⎝∑

�

mx�(t)X

�t + mv(t)V L

t + md(t)Dt − mh(t)Ht

⎞⎠ dt

+⎛⎝∑

�

lx� (t)X�t + lv(t)V L

t + ld (t)Dt − lh(t)Ht

⎞⎠ dBt (4)

and

Dt =∑�

dx� (t)X�

t + dv(t)V Lt + dh(t)Ht , (5)

along with risk premia, �(t), free cash flow growth rate, g(t), and volatilities, �(t) and �H (t),that depend only on time. An implication of g(t) and �(t) depending only on time is that the“price-earnings ratio” for an unlevered asset, yU

t = V Ut /Xt , depends only on time. 12

One can show that the value of a levered asset with debt policy and history satisfying theseproperties is of the additively separable form

V Lt = V U

t +∑�

cx�(t)X�

t + ch(t)Ht

and the PDE, Eq. (1), is of the form 13

∑�

�cx�

�tX� + �ch

�tH + (g − �)

∑�

�cx�X� + 1

2�2∑�

cx��(� − 1)X�

+⎡⎣−

(kh − kddh − (kv + kddv)ch

)H +

∑�

(kx� + kddx

� + (kv + kddv)cx�

)X�

+ (kv + kddv)V U

⎤⎦ ch − rf

⎛⎝∑

�

cx�X� + chH

⎞⎠

= −rf �c

⎡⎣∑

�

dx� X� + dv

⎛⎝V U +

∑�

cx�X� + chH

⎞⎠+ dhH

⎤⎦ ,

with kq = mq − �lq for q ∈ {x, d, v, h}.

12 To prove this, note that Xt drops out of the ratio

V Ut

Xt=∫ T

te∫ st g(�)d� e− ∫ s

t rU� d� ds.

13 Note that many of terms involving V U cancel because of the no arbitrage PDE for V U.


Equating the coefficients of H and X� on each side produces N + 1 ordinary differentialequations with N being the number of powers of X that appear in the debt and history equations:

dcx�

dt+ (g − �)�cx

� + 1

2�2cx

��(� − 1)

+(kx� + kddx

� + (kv + kddv)cx� + (kv + kddv)yU

t ��,1

)ch

−rf cx� + rf �c

(dx� + dvcx

� + dvyUt ��,1

)= 0,

dch

dt−(kh − kddh − (kv + kddv)ch

)ch − rf ch + rf �c

(dvch + dh

)= 0,

with ��,1 a binary variable that takes on the value 1 if � = 1 and 0 otherwise, 14 and with theterminal condition given by

cx�(T ) = ch(T ) = 0.

This system of Riccati equations is easily solved numerically. These equations also are easilyaltered to allow for default and more general tax shields. However, there are large classes of casesthat have closed-form solutions. We explore these below.

2.5. Additively separable assets with closed-form solutions

Suppose that each of the coefficients kd(t), kv(t), kh(t), dv(t), dh(t) are constant and

kx� (t) = kx

� (0) egk� t ,

dx� (t) = dx

� (0) egd� t ,

with the constant growth parameters gk� and gd

� possibly 0. For expositional clarity, we also assumethat the mean and volatility of the free cash flow growth rate, as well as the market price of risk,are constant. That is, g(t, X) = g, �(t, X) = �, and �(t, X) = �. This allows us to express thevalue of an unlevered asset with the growing annuity formula, as in the Gordon Growth Model:

V Ut = Xt

rU − g

(1 − e−(rU−g)(T −t)

). (6)

The Gordon growth assumptions imply that � ln(V Ut )

� ln(Xt )= 1 and that the risk premium on the

unlevered asset rU − rf = �.We could allow g, �, and � to be deterministic functions of time and still achieve solutions

similar to those developed below but at the cost of expressions with confusing sets of integralsin them. As this discussion is about the valuation of tax shields for complex debt policies, andnot about the complexities of valuation in a no-tax setting, we opt for an approach that makes thelatter valuation as uncomplicated as possible.

14 Without loss of generality, and only for notational simplicity, we assume that one of the powers of � is � = 1 if oneof dv , kv , dx

1 , or kx1 is nonzero. Also, note that if �i = 0, we have an exponentially growing constant term. We explore a

special case with this feature later.


Under these assumptions, the system of Riccati equations is solved by

ch(t) = 1 − e−b(T −t)

1 − ch∞ch∞+b/a

e−b(T −t)ch∞, (7)

where 15

a = kv + kddv,

b =√

(kh − kddh + rf (1 − �cdv))2 − 4a�crf dh,

ch∞ =√

b2 + 4a�crf dh − b

2a

and for � = �1, . . . , �N

cx�(t) = rf �cd

x� (t)C1(g

d� ) + rf �cd

v

rU − g��,1

(C1(0) − e−(rU−g)(T −t)C1(r

U − g))

+kx� (t)C2(g

k�) + kddx

� (t)C2(gd� )

+ a

rU − g��,1

(C2(0) − e−(rU−g)(T −t)C2(r

U − g)), (8)

where

C1(z) = 1

1 − ch∞ch∞+b/a

e−b(T −t)

[1 − e−(gx

� −z)(T −t)

gx� − z

−e−b(T −t) ch∞ch∞ + b/a

(1 − e−(gx

� −z−b)(T −t)

gx� − z − b

)],

C2(z) = ch∞1 − ch∞

ch∞+b/ae−b(T −t)

[1 − e−(gx

� −z)(T −t)

gx� − z

−e−b(T −t)

(1 − e−(gx

� −z−b)(T −t)

gx� − z − b

)],

with gx� defined by

gx� = rf (1 − �cd

v) − (kv + kddv)ch∞ + (� − g)� − 12�2�(� − 1).

This closed-form solution can be extended to the case of bankruptcy and more general tax shields.For example, if the instantaneous flow from the tax reduction replaces D in Eqs. (4) and (5), thesolution for gx

� above requires that we replace the first term, rf (1 − �cdv), with the difference

between the default-adjusted short rate and dv .Case 1: Perpetual assets. As T → ∞, Eq. (7) has

ch(t) = ch∞

15 For the tax shield of a finite-lived asset to have a finite value, b, given below, has to be a real number. Also, for historyto be stable, kh − kddh + rf (1 − �cd

v) has to be positive. Throughout the paper, we assume that parameters satisfy thetransversality conditions so that the debt tax shield is finite.


and the N cash flow coefficients given by Eq. (8) simplify to

cx�(t) = kx

� (t)ch∞gx� − gk

�

+ (rf �c + kdch∞)dx� (t)

gx� − gd

�

+ ach∞ + rf �cdv

(rU − g)(rf (1 − �cdv) + � − g)��,1

since, as T → ∞,

C1(z) = ch∞gx� − z

,

C2(z) = 1

gx� − z

.

This implies

V Lt =

(1 + ach∞ + rf �cd

v

rf (1 − �cdv) + � − g

)V U

t +∑�

(kx� (t)ch∞

gx� − gk

�


gx� − gd

�

)X�

t

+ch∞Ht .

Case 2: Perpetual debt as a function of cash flows only. When T → ∞ and dv = dh = 0,ch∞ = 0. In this case, the solution for Case 1 simplifies to

cx�(t) = rf �cd

x� (t)

gx� − gd

�

implying

V Lt = V U

t +∑�

rf �c

rf + (� − g)� − 12�2�(� − 1) − gd

�

dx� (t)X�

t .

Case 3: Debt that is a linear function of asset value plus constant growth. For this special case,

Dt = dx0 (0) egd

0 t + dvV Lt ,

with the sensitivity of debt to asset value constant; that is, dv(t) = dv . The remaining coefficientsare 0. Note that the constant growth rate component in debt, gd

0 , may differ from g, the expectedgrowth rate in cash flows. When dv is 0, debt grows at the constant geometric rate of gd

0 (possibly0). When dx

0 (0) = 0, the debt to asset ratio is constant over the life of the asset. Hence, this policy,as well as the stationary model described in the prior subsection, nests both the Modigliani–Millerand Miles–Ezzell debt policies. (Note that if dv is nonzero, the expected growth rate in debt isinfluenced both by the expected growth rate in V L as well as gd

0 .)This is a case where ch(t) = ch∞ = C2(z) = 0 and the solution reduces to

V Lt = V U

t + cx0 (t) + cx

1 (t)Xt ,

with the values for cx� from Eq. (8) becoming

cx0 (t) = rf �cd

x0 (t)

(1 − e−(rf (1−�cd

v)−gd0 )(T −t)

rf (1 − �cdv) − gd0

),

cx1 (t) = rf �cd

v

rU − g

(1 − e−(rf (1−�cd

v)+�−g)(T −t)

rf (1 − �cdv) + � − g

).


A particularly simple expression exists for a perpetual levered asset in Case 3. Here, the coefficientsabove imply 16

V Lt = V U

t + �c

(rf

rf (1 − �cdv) − gd0

dx0 (t) + rf

rf (1 − �cdv) + � − gdvV U

t

). (9)

Note that when g = gd0 = dv = 0, Eq. (9) is the Modigliani–Miller value,

V Lt = V U

t + �cdx0 (t)

= V Ut + �cDt .

When dv is 0 but gd0 and g are nonzero, we have an extension of the Modigliani–Miller debt

policy that allows for growing debt and free cash flows that are expected to grow. As we will learnin the next section of the paper on the WACC, the initial weighted average cost of capital, usedas a discount rate, does not generate this value as the value of the levered asset when gd

0 �= g.However, there is a simple adjustment to the WACC that generates the correct levered asset value.

When dx0 (t) = 0, debt policy is an extension of the continuous-time Miles and Ezzell debt

policy that allows for cash flows with nonzero expected growth. In this case, Eq. (9) indicates thatthere is a proportional relationship between the value of a levered asset and its otherwise identicalunlevered counterpart:

V Lt = rf + � − g

rf (1 − �cdv) + � − gV U

t

implying

V Lt = V U

t + rf

rf + � − g�cDt .

If gd0 = g, the debt tax shield can be written as a simple weighted average of the tax shields

for the Miles–Ezzell constant leverage ratio debt policy and the extended Modigliani–Miller debtpolicy (with possibly growing debt). In this case, Eq. (9) reduces to

V Lt = V U

t + wt

rf

rf − g�cDt + (1 − wt)

rf

rf + � − g�cDt ,

where the weight

wt = rf − g

rf (1 − �cdv) − g

(1 − dv

Dt/V Lt

).

16 One can also map Dt into V Lt . A small amount of algebraic manipulation reveals

V Lt = V U

t + rf

rf − gd0

�c

(Dt + g − � − gd

0rf (1 − �cdv) + � − g

dvV Ut

).

Thus, the value of the debt tax shield is the sum of the Modigliani–Miller debt tax shield (with constantly growing debt)and a term which may be positive or negative. The sign of the final term in parentheses depends on whether g − �, therisk-adjusted growth rate of the free cash flows, is larger than gd

0 , the growth rate for the nonstochastic debt component.


This weight is monotonically decreasing in dv , the sensitivity of debt to the value of the asset,holding Dt fixed. 17

The Modigliani–Miller debt tax shield, which multiplies wt above, has a smaller value thanits Miles–Ezzell counterpart, which multiplies (1 − wt) above. 18 Hence, for the same initialdebt level, increasing the debt sensitivity coefficient, dv , while holding the initial debt level fixed,reduces the value of the tax shield. This is because the value of the debt tax shield falls when itsrisk increases, other things equal. If the debt sensitivity coefficient, dv , exceeds Dt/V L

t , so thatwt is negative, the leverage ratio will rise as the asset’s value increases and fall when it decreases.In this case, the value of the asset will be below that obtained with the constant leverage ratioMiles–Ezzell debt policy. Conversely, if dv is negative, so that some of the existing debt is retiredwhen the asset’s value rises, 19 and debt is issued when the asset’s value declines, the value of thedebt tax shield will be above the �cDt value proposed by Modigliani and Miller. This confirmsthe intuition in Grinblatt and Titman [9] and suggests that the appropriate discount rate for freecash flows will be a weighted average of the WACC formulas proposed by Miles and Ezzell andModigliani and Miller. For dv > Dt/V L

t , the weight on the Modigliani–Miller WACC must benegative, for dv < 0, it is above 1, and otherwise, it lies between 0 and 1. As we will show later,the weighting on the WACC formulas of Modigliani–Miller and Miles–Ezzell is identical to theweighting of the respective tax shields given here.

The linear debt policy in Case 3 easily extends to discrete time. The linearity implies thatover any discrete interval, the levered asset is a fixed-weight portfolio of an otherwise identicalunlevered asset and a risk-free security. In this case, the values of the levered and unleveredassets are perfectly correlated, as both are linear functions of the cash flow. Solving the differenceequations that generate the no arbitrage value of the levered asset in an analogous manner yieldsthe discrete time analogue to Eq. (9) 20 :

V Lt = V U

t + �c

(rf

rf (1 − �cdv) − gd0

dx0 (t)

+ rf (1 + rf + �)

(1 + rf )(rf (1 − �cdv) + � − g) − �rf �cdvdvV U

t

).

This valuation solution nests both the discrete-time Modigliani–Miller and Miles–Ezzell debtpolicies as special cases.

The discrete case valuation formula, provided above, applies only to an infinitely-lived asset.A similar approach generates a discrete time closed-form solution for a finite-lived asset. It isomitted for the sake of brevity.

17 To see this, it is necessary to obtain an equation for the weight without V Lt . This is accomplished by substituting the

former equation into the latter and solving for wt .18 This and the statements that follow from it assume that �, the risk premium for the free cash flow, is positive. If the

free cash flow has a negative risk premium, the Miles–Ezzell value exceeds the Modigliani–Miller value.19 This debt paydown pattern has been estimated in empirical work by Kaplan and Stein [23].20 Simple algebraic manipulation indicates that the mapping from Dt to VL associated with the equation below is

given by

V Lt = V U

t + rf

rf − gd0

�c

(Dt + (1 + rf )(g − � − gd

0 ) + �(rf − gd0 )

(1 + rf )(rf (1 − �cdv) + � − g) − �rf �cdvdvV U

t

).


3. The weighted average cost of capital and tax-adjusted discount rates

For finance practitioners, discounting expected free cash flows at a tax-adjusted discount rateis the most popular way to value an asset. This section studies the relation between this discountrate and the after-tax weighted average cost of capital. It develops formulas for these discountrates for a variety of debt policies and shows when and why naive application of the weightedaverage cost of capital as the appropriate discount rate can generate erroneous valuations.

We take the perspective of an investor at date 0. This investor would like to know the discountrate, applied to expected future free cash flows, that generates V L

0 . Our analysis will show thatthis discount rate is rarely the WACC computed at date 0. Before we do this, it is important tostudy the WACC and how it evolves through time.

3.1. Risk, expected return, and the WACC

In continuous time, the weighted average cost of capital of an asset is defined to be the asset’sinstantaneous expected return, rL, less the return component due to the debt tax shield 21 :

WACCt = rLt − rf �c

Dt

V Lt

. (10)

A levered asset’s date t instantaneous expected return, equivalent to its “pre-tax weighted averagecost of capital,” is defined by 22

rLt dt = Et (dV L

t ) + Xt dt + rf �cDt dt

V Lt

. (11)

This expected return has three components: the expected “capital gain,” the free cash flow, andthe cash flow from the debt tax shield. 23 One can readily show from the no arbitrage conditionthat

rLt = rf +

(� ln V L

t

�Xt

+ �H (t, Xt , Dt , VLt )

� ln V Lt

�Ht

)�(t, Xt )Xt .

Combining this equation with Eq. (10) provides a direct formula for computing a WACC giventhe value of the levered asset:

WACCt = rf

(1 − �c

Dt

V Lt

)+(

� ln V Lt

�Xt

+ �H (t, Xt , Dt , VLt )

� ln V Lt

�Ht

)�(t, Xt )Xt . (12)

Eq. (12) is a generalization of the Modigliani–Miller adjusted cost of capital formula. It isconvenient for computing the WACC given the extensive closed-form solutions derived in the last

21 This is the same formula used by practitioners and found in textbooks, as the levered asset’s expected return is aportfolio-weighted average of the pre-tax expected returns of the (assumed risk-free) debt and equity financing the asset.

22 This expected return is the appropriate discount rate for the “capital cash flow stream,” which is the net payout to allcash flow claimants. See Ruback [20] for a lucid discussion of the advantages of this approach.

23 As mentioned earlier, the latter flow must be paid out to maintain the same investment policy and capital gainsappreciation as an otherwise identical unlevered asset.


section. For example, in the case of constant debt for a finite-lived asset with zero expected growthand a constant risk premium for free cash flows,

� ln V L

�H= 0

and

� ln V Lt

�Xt

= V Ut

V Lt

1

Xt

.

In this case, the levered asset has a value of

V Lt = V U

t + �cDt (1 − e−rf (T −t)),

and the formula for the WACC reduces to

WACCt = rU(

1 − �c

Dt

V Lt

)+ �c�

Dt

V Lt

e−rf (T −t).

Note that the second term is decreasing in T and converges to 0 for perpetual assets. Hence, thisformula generates a larger WACC than that generated by the Modigliani and Miller formula forperpetual assets.

It is also possible to write down a differential equation for the WACC. Substituting the expectedreturn formula, Eq. (11), into Eq. (10) implies

WACCt dt = Et

[dV L

t

]+ Xt dt

V Lt

. (13)

Applying Ito’s lemma to this equation, we find that the WACC satisfies the PDE

WACC × V L =(

�V L

�t+ gX

�V L

�X+ 1

2�2X2 �2

V L

�X2+ �h �V L

�H+ 1

2�2

h

�2V L

�H 2

+��h

�2V L

�H�X

)V L + X.

To understand the relation between the WACC and stochastic discount rates for free cash flows,observe that t , the date t stochastic instantaneous discount rate for Xt that generates the valueof the levered asset, satisfies the stochastic integral equation

V Lt =

∫ ∞

t

Et

[e− ∫ s

t �d�Xs

]ds.

By the Feynman–Kac theorem, any t that satisfies this stochastic integral equation also satisfiesthe PDE

V L =(

�V L

�t+ gX

�V L

�X+ 1

2�2X2 �2

V L

�X2+ �h �V L

�H+ 1

2�2

h

�2V L

�H 2+ ��h

�2V L

�H�X

)V L

+X.


Since the PDE that has to solve is identical to the PDE that the WACC solves (by Ito’s Lemma),the following proposition must hold 24 :

Proposition 1. TheWACC is identical to the stochastic instantaneous discount rate that generatesthe levered asset value when applied to the free cash flows.

This insight is not very useful for capital budgeting practitioners. As Eq. (12) indicates, theWACC at a future date depends on the state variables, X and H, at that date. Hence, future WACCsare generally stochastic when viewed from date 0, the relevant date of the valuation.

3.2. The appropriate tax-adjusted discount rates

Even though the WACC is generally stochastic, it may be that some construct related to theWACC can be used to discount expected future free cash flows to date 0 in a manner that accountsfor the tax shield. This subsection explores this issue. We first begin by analyzing a discount rate,known at date 0, that translates the expected free cash flow at date s into its value an instant earlier.We call this the “tax-adjusted forward rate.” Once having developed an understanding of whatthis tax-adjusted forward rate is, and how to compute it, we prove that the initial WACC is anappropriate tax-adjusted discount rate whenever the term structure of tax-adjusted forward rates isflat. These forward rates, while computable, are fairly impractical for capital budgeting purposes.However, they are consistent with a single tax-adjusted discount rate for free cash flows—thetax-adjusted hurdle rate for the IRR—which generates the same present value. Moreover, whenthe unlevered assets are perpetual and the Gordon-Growth assumptions apply, this hurdle rate,denoted ∗, is easily obtained with a simple formula.

Define date 0’s tax-adjusted forward discount rate for cash flows at date s, fs , by

fs ds = E0[dV L

s + Xs ds]

E0[V L

s

] . (14)

This is clearly an appropriate discount rate. It is known at date 0, and the recursive relationshipexpressed in Eq. (14), applied iteratively, implies

V L0 =

∫ ∞

0e− ∫ t

0 fs dsE0 [Xt ] dt. (15)

How does this series of forward rates relate to the WACC? At most horizons, the comparisonis meaningless because the future WACCs are stochastic when viewed from date 0. As a generalmatter, the ratio of date 0 expectations in Eq. (14), used to compute the date s forward rate, differsfrom the date 0 expectation of WACCs . Moreover, the expectations in Eq. (14), while obtainable,do not lend themselves to simple expressions.

Despite this, developing an understanding of tax-adjusted forward rates is useful for under-standing when the initial WACC can be used for discounting. Trivially, fs converges to WACC0 ass approaches 0. Because f0 = WACC0, and fs is an appropriate instantaneous discount rate fordate s cash flows, it follows that whenever fs = f0 for all s, WACC0 is an appropriate discountrate for free cash flows.

24 Obviously, there is no reason for the boundary conditions to differ or for either of the partial differential equations tobe ill-behaved.


Proposition 2. Whenever the term structure of tax-adjusted forward rates is flat (fs = f0, ∀s),the WACC is an appropriate tax-adjusted discount rate for free cash flows.

Two cases where this situation arises are the no-growth Modigliani–Miller model and the Miles–Ezzell model (with or without cash flow growth). The only other situation where the forward termstructure is flat is when debt policy is any weighted average of debt policies in these two models,but only for g = gd

0 . We explore this shortly.The rarity of an equivalence between the initial WACC or expected future WACCs and the

corresponding forward rates should not be surprising. Although Eqs. (13) and (14) look similar,Jensen’s inequality alone prevents the expectation of the former from equalling the latter whenV L

t , as well as the overall ratio in Eq. (13), are stochastic.Date 0’s constant tax-adjusted discount rate, ∗, is defined by the following equation

V L0 =

∫ ∞

0e−∗tE0[Xt ] dt.

In cases where the free cash flows have perpetual constant growth g, this tax-adjusted discount(hurdle) rate is most easily computed as

∗ = g + X0

V L0

. (16)

Eq. (16), an algebraic manipulation of the growing perpetuity formula, allows us to obtain ∗from the formulas for V L

0 developed in the prior section. 25

3.3. Classes of cases with easy numerical solutions

Recall from the last section that the values of assets with time-independent cash flows anddebt policy, as well as assets with additively separable history diffusion and debt equations couldeasily be obtained numerically. In the former class of cases, the fundamental valuation equationreduces to an ordinary differential equation in X. In the latter class, it reduces to a system of Riccatiequations. In these cases, the WACC and appropriate discount rate, ∗, are similarly solved. Just asnumerical solutions for V L are easily obtained, so too are solutions for the WACC using Eq. (12).For ∗, the formula in Eq. (16) generates the discount rate directly from the numerically solvedV L. 26 Similarly, the forward rates are the solutions to ordinary differential equations which canbe solved numerically. 27

Obviously, it is more illuminating to analyze closed-form solutions for these interesting vari-ables. We turn our attention to this next.

25 With a finite-lived asset, ∗ is easily identified implicitly as the parameter that solves

V L0 = X0

∗ − g

(1 − e−(∗−g)(T −t)

).

26 For finite-lived assets, this discount rate can be easily solved implicitly, as described in the prior footnote.27 The ODEs are available upon request.


3.4. Additively separable assets with closed-form solutions

Consider, as in the last section, the case where history follows the diffusion

dHt = (mx(t)Xt + mdDt + mvV Lt − mhHt)dt + (lx(t)Xt + ldDt + lvV L

t − lhHt )dBt

and debt policy has the functional form

Dt =∑�

dx� (t)X�

t + dvV Lt + dhHt .

Recall that g, �, and � are constant, dx� (t) = dx

� (0) egd� t , mx

�(t) − �lx� (t) = kx� (t) = kx

� (0) egk� t ,

and T = ∞.Here, the partial derivatives in Eq. (12) have closed-form solutions, allowing us to express

the WACC as an explicit function of the free cash flow, history state variable, and leverage ratio,Dt/Vt , as follows:

WACCt = rf

(1 − �c

Dt

V Lt

)+ 1/(rf + � − g) +∑

� cx�(t)�X�−1

t + ch(t)�H

V Ut +∑

� cx�(t)X�

t + ch(t)Ht

�Xt, (17)

where the c coefficients are explicitly given in the prior section of the paper and with

�H =∑�

lx� (t)Xt + ldDt + lvV Lt − (lh + �)Ht .

Case 1: Perpetual assets. Here, Eq. (17) simplifies to the same expression, but with

ch(t) = ch∞

and

cx�(t) = kx

� (t)ch∞gx� − gk

�


gx� − gd

�

+ (kv + kddv)ch∞ + rf �cdv

(rU − g)(rf (1 − �cdv) + � − g)��,1,

where the constants gx� and ch∞ are given in the previous section of the paper. In contrast, the

constant tax-adjusted discount rate is

∗ = g + X0∑� cx

�(0)X�0 + ch∞H0

.

Case 2: Perpetual debt as a function of cash flows only. Consider, as in the last section, the casewhere

Dt =∑�

dx� (t)X�

t ,

g, �, and � are constant, dx� (t) = dx

� (0) egd� t and T = ∞. For this case, we showed that

V Lt = V U

t +∑�

rf �c

rf + (� − g)� − 12�2�(� − 1) − gd

�

dx� (t)X�

t .


This valuation solution simplifies Eq. (17) to

WACCt = rf

(1 − �c

Dt

V Lt

)+ 1/(rf + � − g) +∑

� cx�(t)�X�−1

t

V Ut +∑

� cx�(t)X�

t

�Xt,

where

cx�(t) = rf �cd

x� (t)

rf + (� − g)� − 12�2�(� − 1) − gd

�

,

while the constant tax-adjusted discount rate

∗ = g + X0

V U0 +∑

�rf �c

rf +(�−g)�− 12 �2�(�−1)−gd

�dx� (0)X�

0

.

This happens to be a case where the presentation of the term structure of tax-adjusted forwardrates computed from the date 0 valuation date will not significantly lengthen the paper. To do this,we need to take date 0 expectations of V L

s . Given the lognormal process for Xs , the formula forthe conditional �th moment of Xs is

E0

[X�

s

]= X�

0 e�(g− 12 �2)s+ 1

2 �2�2s ,

which can be substituted into the expected value of Eq. (14). Thus,

E0[V Ls ] = V U

0 egs +∑�

�crf dx�

X�0 e�(g− 1

2 �2)s+ 12 �2�2s

rf + (� − g)� − 12�2�(� − 1)

.

It follows that

dE0[V Ls ]

ds= V U

0 g egs +∑�

�crf dx�

(�(g − 12�2) + 1

2�2�2)X�0 e�(g− 1

2 �2)s+ 12 �2�2s

rf + (� − g)� − 12�2�(� − 1)

.

We now have all the ingredients to compute the date s tax-adjusted forward rate. It is given by theformula:

fs =V U

0 (rf + �) egs +∑� �crf dx

�

(�(g − 12�2) + 1

2�2�2)X�0 e�(g− 1

2 �2)s+ 12 �2�2s

rf + (� − g)� − 12�2�(� − 1)

V U0 egs +∑

� �crf dx�

X�0 e�(g− 1

2 �2)s+ 12 �2�2s

rf + (� − g)� − 12�2�(� − 1)

.

It is easily verified that WACC0 = f0, but obviously, the date s WACC depends on Xs and thuscannot be identical to fs . Moreover, the date s forward rate is not the date 0 expectation of thedate s WACC, irrespective of whether expectations are taken with respect to the actual probabilitydensity function or the probability density function generated by the risk-neutral measure.

Case 3: Perpetual debt that is a linear function of asset value plus constant growth. Recall thatfor this special case, explored in the last section, debt policy is given by

Dt = dx0 (0) egd

0 t + dvV Lt


and a perpetual levered asset has a particularly simple functional form for its valuation:

V Lt = V U

t + �c

(rf

rf (1 − �cdv) − gd0

dx0 (t) + rf

rf (1 − �cdv) + � − gdvV U

t

).

This valuation solution simplifies Eq. (17) to

WACCt = rf

(1 − �c

Dt

V Lt

)+ �Xt

(rf (1 − �cdv) + � − g)V Lt

,

with V Lt given by the equation immediately above, while the appropriate tax-adjusted constant

discount rate, ∗, is given by

∗ = g + X0

V U0 + �c

(rf

rf (1 − �cdv) − gd0

dx0 (t) + rf

rf (1 − �cdv) + � − gdvV U

0

) .

Using Eq. (14), the simple valuation formula for this case can be used to show that the tax-adjusted forward discount rate is given by the weighted average

fs = 0(s)

0(s) + 1(s)gd

0 + 1(s)

0(s) + 1(s)(rf (1 − �cd

v) + �),

where the coefficients

0(s) = rf �cdx0 (0)

rf (1 − �cdv) − gd0

egd0 s ,

1(s) = X0

rf (1 − �cdv) + � − gegs

and

V L0 = 0(0) + 1(0).

It is possible to compare forward rates, WACCs, and appropriate constant tax-adjusted discountrates here but the discussion is more illuminating if we focus on several special cases of thisexample.

Case 3a: Constantly growing debt with no stochastic component and no expected cash flowgrowth. This extension of the Modigliani–Miller debt policy accounts for the possibility of growingdebt, g = dv = 0, and rf > gd

0 > 0. The 0(s) and 1(s) coefficients above simplify to

0(s) = rf �cD0

rf − gd0

egd0 s = �0 egd

0 s ,

1(s) = X0

rf + �= V U

0

implying

fs = �0 egd0 sgd

0

V U0 + �0 egd

0 s+ V U

0

V U0 + �0 egd

0 srU.


The initial WACC, which is identical to f0, is

WACC0 = �0

V L0

gd0 + V U

0

V L0

rU

= rU

(1 − �c

D0

V L0

)− �

rf − gd0

gd0 �c

D0

V L0

.

However, the tax-adjusted constant discount rate is

∗ = X0

V L0

= rU V U0

V L0

= rU

(1 − rf

rf − gd0

�c

D0

V L0

),

which the initial WACC exceeds by the amount

�0

V L0

gd0 = rf

rf − gd0

gd0 �c

D0

V L0

.

Case 3b: Modigliani–Miller static debt policy with constant expected growth in cash flows.Here we assume that g > 0, but that dv = gd

0 = 0. In this case

0(s) = �cD0,

1(s) = X0

rf + � − gegs = V U

0 egs

implying

fs = V U0

V L0

egsrU.

The initial WACC, which is identical to f0, is

WACC0 = V U0

V L0

rU = rU

(1 − �c

D0

V L0

).

However, the tax-adjusted constant discount rate is

∗ = X0

V L0

+ g = V U0

V L0

(rU − g) + g

= V U0

V L0

rU + g�c

D0

V L0

,

which exceeds the initial WACC by g�cD0V L

0. Like the other cases above, using the WACC as a

discount rate here can lead to serious valuation errors. For example, a company with a constantamount of debt, a current free cash flow of $100 million, a perpetual cash flow growth rate of 5%,a 30% corporate tax rate, and a 50% debt to asset ratio can plausibly have a true present value of$2 billion. This present value would stem from a properly computed tax-adjusted discount rateof 10%. The WACC in this case, 9.25%, leads to a valuation of more than $2.35 billion. What isreally striking here is the degree to which the tax shield is misvalued. The true value of the entiretax shield is $300 million which is smaller than the valuation error arising from using the WACCas the discount rate for free cash flows.


Case 3c: The Modigliani–Miller model with equal growth in debt and expected cash flows. Thelast two cases illustrated that the WACC was not an appropriate tax-adjusted constant discountrate. This is because it changed over time. In Case 3a, the initial WACC was too large because itwas expected to decrease. In Case 3b, the initial WACC was too small because it was expected toincrease. It is natural to think that if dv = 0 but gd

0 = g > 0, so that debt growth keeps pace withthe expected growth of the cash flows, the initial WACC will still work. This indeed is the case.To prove this, note that here

WACC0 = f0 = �0

V L0

gd0 + V U

0

V L0

rU = rU

(1 − �c

D0

V L0

)− �

rf − gd0

gd0 �c

D0

V L0

.

Thus, WACC0 = ∗ whenever

�0

V L0

gd0 + V U

0

V L0

rU = g + (rU − g)V U0

V L0

,

which only occurs when gd0 = g.

This happens to be a case in which the forward rate is independent of s. By Proposition 2, theinitial WACC works as an appropriate tax-adjusted discount rate under this condition.

Case 3d: A combination of the extended Modigliani–Miller and Miles–Ezzell debt models withequal growth in debt and expected cash flow. Here, we generalize Case 3c and once again showthat the initial WACC is an appropriate discount rate for free cash flows. In this example, gd

0 = g

but dv may be nonzero. These assumptions imply:0(s)

0(s) + 1(s)= 1 − X0

V L0

× 1

rf (1 − �cdv) + � − g,

1(s)

0(s) + 1(s)= X0

V L0

× 1

rf (1 − �cdv) + � − g.

To compute fs , the former ratio multiplies g, while the latter multiplies rf (1 − �cdv) + �, and

then the two products are summed. Thus,

fs = f = ∗ = WACC0 = g + X0

V L0

.

When gd0 �= g, fs depends on s. Hence, within Case 3, it is only by imposing the condition gd

0 = g

that one can use the initial WACC as an appropriate discount rate. This discount rate can be thoughtof as a weighted average of the WACC from a Miles and Ezzell debt policy at an initial debt ofD0 and the WACC from a Modigliani and Miller debt policy at an initial debt of D0. Specifically,denote the WACCs of Modigliani–Miller and Miles–Ezzell as

WACCMM0 = rU

(1 − �c

D0

V L0

)− g�

rf − g�c

D0

V L0

,

WACCME0 = rU − �crf

D0

V L0

,

respectively. Then the discount rate given above

WACC0 = ∗ = f = w0WACCMM0 + (1 − w0)WACCME

0 ,


where

w0 = rf − g

rf (1 − �cdv) − g

(1 − dv

D0/V L0

).

Case 3e: The growth extension of the Miles–Ezzell model. Consider the case with dv = D0/V0,dx

0 (0) = 0. This is a continuous-time version of the Miles–Ezzell model, but with the extensionof expected growth in cash flows. Here,

V Ut = Xt

rf + � − g

and

V Lt = V U

t + rf

rf + � − g�cDt .

In this case, since 0(s) is 0, the date s forward rate is independent of s, which has the value

f = ∗ = rU − rf �c

D0

V L0

.

This tax-adjusted discount rate is identical to the weighted average cost of capital.To understand this from another perspective, recognize that everything is stationary here, and

that the flow from the debt tax shield, being proportional to the free cash flow, shares its discountrate, so that

rL = rf + �.

The WACC, from Eq. (10), is therefore the constant

WACCt = rf + � − rf �c

D

V L ,

where by assumption, D/V L is the same at all points in time. Note that the growth rate of thecash flows never appears when the WACC is written as a function of the leverage ratio, but that itaffects V L above.

3.5. Discussion

To develop an intuitive understanding of the results above, it is useful to first review whydiscounting free cash flows at the initial after-tax weighted average cost of capital sometimesaccounts for the value of the debt tax shield. Let us begin with the perpetual level debt model ofModigliani and Miller. Because the market value of the debt financing never changes in this model,it is useful to think of the Modigliani and Miller analysis as the valuation of the debt tax shieldof a zero beta debt strategy. In such a model, the familiar equation V L = V U + �cD reflects theseparate valuations of the two components of the asset: the free cash flow stream, with stochasticvalue Xt at date t, and the stream of cash flows from the interest-based tax shield, Drf �c, whichis constant and identical at every date. Note that the gross (as opposed to net) pre-tax weightedaverage cost of capital, which is also the gross expected return on the levered asset, is

1 + rL = 1 + (D/V L)rf + (E/V L)rE.


This can be viewed as the expected cash flow per dollar of assets to investors who buy an asset,hold it for an instant, and then liquidate it. Both the asset value and the flow include the debt taxshield. Hence, the gross after-tax WACC

1 + (D/V L)rf (1 − �c) + (E/V L)rE

is just the expected flow to investors per dollar of levered assets, less the expected flow per dollarof assets from the debt tax shield (D/V L)rf �c. This net flow is identical to the free cash flowper dollar of levered assets. Viewed with the arrow of time in reverse, this insight implies thatif we discount the free cash flow at the after-tax WACC, we end up back where we started, withthe one dollar value of the assets, including that component of value generated by the debt taxshield.

It is easy to see that this insight about the zero beta debt strategy does not easily extend to thecase of an asset with a value that tends to grow. Here, the value of the unlevered asset is affectedby the growth rate, but the debt tax shield, having a perpetual value of �cD, is not affected by it.Because the value of such a constant debt tax shield tends to decline over time as a proportion oftotal asset value, the initial after-tax weighted average cost of capital is an inappropriate discountrate for all future cash flows. Essentially, the weighted average cost of capital is changing overtime. In this simple case, we learned that there is a single discount rate that can be used to discountall future free cash flows: the sum of the initial after-tax WACC and g�cD/V L, the latter beingthe product of the growth rate of the free cash flows, the tax rate, and the debt to value ratio. Fora firm with a 4% growth rate, and a 50% tax rate and leverage, this alone represents a 100 basispoint error in the discount rate, even given the true unlevered cost of capital. We can see thatif the risk of the levered asset is not expected to change, as in the Modigliani–Miller model ofdebt adjustment, which assumes no growth, discounting the expected free cash flow stream at theinitial WACC generates the value of the levered assets, including the debt tax shield.

The other case where the risk of the levered asset is not expected to change arises in an extensionof the Miles and Ezzell [16] model. Their case is one where the debt level is adjusted over time tomaintain a constant debt to value ratio. This is a positive beta debt strategy even though the debt,at issue, is risk-free. The perpetually constant WACC is an appropriate discount rate in this case,but the valuation is never out of peril if the actual debt policy associated with the asset differs fromthe policy implicit in the formulas used for the valuation. For example, if the unlevered asset’srisk premium is twice the risk-free rate, the Miles–Ezzell tax shield is a mere one-third that ofthe Modigliani–Miller debt tax shield for the same debt level. If the risk-free rate is 4%, the biasin the WACC from applying a Miles–Ezzell formula to a Modigliani–Miller debt policy becomes200 basis points, even if the unlevered cost of capital for the asset is estimated perfectly.

These problems can become much more severe when the unlevered cost of capital is fromcomparison assets with debt policies that are also mismatched to the formulas. For example,an asset with a nonstochastic trend to grow its debt but pay it down if the cash flows turn outto be surprisingly good, has a very low WACC other things equal. Unless this is recognized,unlevering such an asset with a standard formula tends to produce too low an unlevered cost ofcapital. Similarly, obtaining the WACC of such an asset at a target debt level using the standardformulas tends to produce too large a WACC. In contrast, consider an asset for which debt has anonstochastic retirement trend, but for which debt tends to increase rapidly when cash flows aresurprisingly good. Applying the standard formulas to such an asset tends to generate unleveredcosts of capital that are too high given a known WACC and a WACC at a target debt level that istoo low given the asset’s unlevered cost of capital. If the former asset has a known WACC and is


used to generate the WACC for the latter asset (or vice versa), the bias can easily run to 500 basispoints or more. 28

4. Levering and unlevering equity betas

The relationship between equity betas and unlevered asset betas is critical for valuation. Equitybetas, unlike asset betas, are observed. Hence, to obtain the necessary discount rates, valuationsanalyze the traded equity of assets that are deemed similar to the asset being valued. The problem isthat the risk of the traded equity of comparison assets is affected by the leverage policy associatedwith the comparison asset. It is generally necessary to undo the leverage-induced distortion on theequity beta of the comparison asset(s) in order to obtain the critical valuation inputs: the unleveredasset beta or unlevered cost of capital. This section explores how to do this.

The return on an unlevered asset is given by

dV U + Xt dt

V U =(

rf + �� ln V U

� ln X

)dt + � ln V U

� ln X� dB,

while the return on a levered asset is given by

dV L + X dt + rf �cD dt

V L =(

rf +(

� ln V L

�X+ �H

� ln V L

�H

)�X

)dt

+(

� ln V L

�X+ �H

� ln V L

�H

)�X dB.

The ratio of the terms that multiply dB in the two equations above represent the effect of leverageon volatility. With risk-free debt, the proportional effect on beta is the same. Moreover, with risk-free debt, the equity beta of a levered asset always is 1+D/E times the beta of the underlying asset(including the asset component from the tax shield). 29 It follows that the formula for leveringand unlevering equity betas involves multiplying 1 + D/E by the ratio of the terms that multiplydB above. That is,

�Et =

(1 + Dt

ELt

) � ln V L

�X+ �H

� ln V L

�H

� ln V U

�X

�Ut ,

with the derivatives taken at date t.The above equation suggests that whenever there is a closed-form solution for the levered asset

value, there is a closed-form solution for the equity beta formula, obtainable after first takingpartial derivatives. For example, the extension of the Modigliani–Miller debt policy with g �= 0and gd

0 �= 0 has

� ln V L

�H= 0

and

� ln V L

�X= V U

V L

� ln V U

�X,

28 Moreover, this analysis assumes that the WACC is an appropriate discount rate, which it rarely is.29 This is just an inversion of the portfolio formula that generates the beta of the levered asset as a portfolio-weighted

average of the beta of its debt (0) and equity (�E ).


which implies

�Et =

[1 +

(1 − �c

rf

rf − gd0

)Dt

ELt

]�U

t .

When gd0 is nonzero, this formula differs from the leveraging–unleveraging formula proposed by

Hamada [10] for the Modigliani and Miller debt policy. It also differs from this formula wheng = gd

0 = 0 but the asset has a finite life. In the latter case,

� ln V L

�H= 0

and

� ln V L

�X= V U

V L

� ln V U

�X.

However, because the finite horizon debt tax shield has a different value than the perpetual debttax shield, the formula reduces to

�Et =

[1 +

(1 − �c(1 − e−rf (T −t))

) Dt

ELt

]�U

t .

In contrast, the extension of the Miles–Ezzell policy with g �= 0 has

� ln V L

�H= 0

and

� ln V L

�X= � ln V U

�X= 1/X

implying

�Et =

(1 + Dt

ELt

)�U

t .

This is identical to the formula proposed by Miles and Ezzell and it applies to both an infinitely-lived and finite-lived asset (in contrast to the Hamada formula for the Modigliani–Miller debtpolicy).

For a perpetual asset that is a hybrid of the Modigliani–Miller and Miles–Ezzell debt policies,as given in Case 3 of the prior sections of the paper, the equity beta formula is

�Et =

[1 +

(1 + �c

(dv

Dt/Et

− 1

)rf

rf (1 − �cdv) − gd0

)Dt

Et

]�U

t .

The formula above is a weighted average of the beta leveraging formulas of Hamada/Modigliani–Miller and Miles–Ezzell. That is

�Et =

[wt

(1 +

(1 − �c

rf

rf − gd0

)Dt

ELt

)+ (1 − wt)

(1 + Dt

Et

)]�U

t ,


with the weight on the Hamada/Modigliani–Miller formula

wt = rf − gd0

rf (1 − �cdv) − gd0

(1 − dv

Dt/V Lt

).

There are several additional cases with simple closed-form solutions for the equity beta. Theseparallel cases 1 and 2 in the prior two sections of the paper. Since these involve mere substitutionsand elementary calculus, we omit them for the sake of brevity.

Finally, we note that since equity risk premia are proportional to equity betas, the formulas forleveraging and unleveraging equity risk premia are the same as those above, with the levered andunlevered equity risk premia substituting for their respective betas above.

5. Conclusion

This paper has undertaken a comprehensive valuation of debt tax shields. In as many casesas possible, we offered closed-form solutions for the values of levered assets and the associateddebt tax shields. Our approach for obtaining these present values was the APV approach. Thetax-adjusted discount rates that generate the present values were reverse engineered, in that weneeded to use the present values to generate closed-form solutions for the discount rates.

The examples explored in this paper are particularly useful in that many dynamic debt policiescan be thought of as fitting into one of these examples. The examples demonstrated that it ispossible to develop intuition from polar cases so that a manager can heuristically assess how hisdiscount rate and debt tax shield value will change, given the dynamic nature of the policy. Forinstance, firms with debt policies that sluggishly but positively react to changes in the value of thefirm’s unlevered assets might be expected to have a debt tax shield with a value that lies somewherebetween the Modigliani–Miller and Miles–Ezzell values. The appropriate tax-adjusted constantdiscount rate for the future cash flows of the unlevered assets, as well as the formulas for leveringand unlevering equity betas will also lie between the values given by the polar cases. Moreover,when firms engage in what we term “negative beta debt policies,” paying down debt as the cashflow prospects brighten, and vice versa, the formulas for the debt tax shield, discount rate, andlevered equity betas are again weighted averages of the two polar cases, with a negative weighton the formula associated with the Miles–Ezzell positive-beta debt strategy and a weight above 1on the Modigliani–Miller zero-beta debt strategy formula.

The comprehensive treatment of debt tax shields presented here is essential for practitioners.Confusion has proliferated because the formulas that previously had been developed for thesimplest of cases are generally treated as black boxes without a clear understanding of where theycome from. It is rare when pedagogy appropriately links the formulas for levering and unleveringbetas to the value of the tax shield. The Miles–Ezzell valuation can easily be 1/3 the valuationusing the Modigliani–Miller approach. It is quite common, however, to observe both students andfinance professors mix the Miles and Ezzell formula for leveraging and unleveraging equity withModigliani–Miller inputs, thinking that they are getting a valuation of �cD for the debt tax shield.In properly linking the values of tax shields for different debt policies to discount rates and layingout the theory behind this linkage, we hope to remedy some of this confusion.

This paper is important, however, not just for those doing corporate valuations, but for thosedoing research on capital structure and bankruptcy costs. In particular, it provides a set of bench-marks for the impact of debt on asset values in a market that is frictionless, except for taxes.Empirical research by Graham [6] and by Kemsley and Nissim [14] indicates that for the average


US firm, debt tax shields in the US are about the size of �cD. Clearly, more research that describesthe cross-sectional variation in this estimate is coming, and is warranted. To properly estimatebankruptcy costs from extensions of this research, it is critical to know the value of the debttax shield in a market where the only friction is taxes. We believe that research in this area hasbeen hindered by a lack of understanding of benchmark valuations. By plugging this hole in theliterature, we hope to stimulate additional research.

Acknowledgments

The authors thank the UCLA Academic Senate and the Harold Price Center for EntrepreneurialStudies for financial support and seminar participants at the University of Arizona, the Universityof Kentucky, and the Financial Management Association meeting for comments on earlier drafts.

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